I haven’t written anything here in a while, but hope to write more regularly now that the semester is over—I have a seriesoncombinatorialproofs to finish up, some books to review, and a few other things planned. But to ease back into things, here’s a little puzzle for you. Recall that the Fibonacci numbers are defined by

.

Can you figure out a way to prove the following cute theorem?

If evenly divides , then evenly divides .

(Incidentally, the existence of this theorem constitutes good evidence that the “correct” definition of is , not .)

Nope, but if you know of such a proof I’d love to hear it! One of the proofs definitely uses induction, and can probably be formulated in terms of dominos, though I haven’t thought about the details. The other one is more elementary.

I would also like to see a proof involving monimos, dominos, and geronimos.

Reblogged this on nebusresearch and commented:
I have not, as far as I remember, encountered this theorem before. And for the time I’ve had to think about it I realize I’ve got no idea how to prove it. However, it’s a neat little result that makes me smile to hear about, and theorems that bring smiles are certainly worth sharing.